Nonlinear Dendritic Coincidence Detection for Supervised Learning APREPRINT

NONLINEAR DENDRITIC COINCIDENCE DETECTION FOR SUPERVISED LEARNING APREPRINT Fabian Schubert Claudius Gros Institute for Theoretical Physics Institute for Theoretical Physics Goethe University Frankfurt Goethe University Frankfurt Frankfurt am Main Frankfurt am Main [email protected] [email protected] July 13, 2021 ABSTRACT Cortical pyramidal neurons have a complex dendritic anatomy, whose function is an active research field. In particular, the segregation between its soma and the apical dendritic tree is believed to play an active role in processing feed-forward sensory information and top-down or feedback signals. In this work, we use a simple two-compartment model accounting for the nonlinear interactions between basal and apical input streams and show that standard unsupervised Hebbian learning rules in the basal compartment allow the neuron to align the feed-forward basal input with the top-down target signal received by the apical compartment. We show that this learning process, termed coincidence detection, is robust against strong distractions in the basal input space and demonstrate its effectiveness in a linear classification task. Keywords Dendrites · Pyramidal Neuron · Plasticity · Coincidence Detection · Supervised Learning 1 Introduction In recent years, a growing body of research has addressed the functional implications of the distinct physiology and anatomy of cortical pyramidal neurons [Spruston, 2008, Hay et al., 2011, Ramaswamy and Markram, 2015]. In particular, on the theoretical side, we saw a paradigm shift from treating neurons as point-like electrical structures towards embracing the entire dendritic structure [Larkum et al., 2009, Poirazi, 2009, Shai et al., 2015]. This was mostly due to the fact that experimental work uncovered dynamical properties of pyramidal neuronal cells that simply could not be accounted for by point models [Spruston et al., 1995, Häusser et al., 2000]. An important finding is that the apical dendritic tree of cortical pyramidal neurons can act as a separate nonlinear arXiv:2107.05336v1 [q-bio.NC] 12 Jul 2021 synaptic integration zone [Spruston, 2008, Branco and Häusser, 2011]. Under certain conditions, a dendritic Ca2+ spike can be elicited that propagates towards the soma, causing rapid, bursting spiking activity. One of the cases in which dendritic spiking can occur was termed ‘backpropagation-activated Ca2+ spike firing’ (‘BAC firing’): A single somatic spike can backpropagate towards the apical spike initiation zone, in turn significantly facilitating the initiation of a dendritic spike [Stuart and Häusser, 2001, Spruston, 2008, Larkum, 2013]. This reciprocal coupling is believed to act as a form of coincidence detection: If apical and basal synaptic input co-occurs, the neuron can respond with a rapid burst of spiking activity. The firing rate of these temporal bursts exceeds the firing rate that is maximally achievable under basal synaptic input alone, therefore representing a form of temporal coincidence detection between apical and basal input. Naturally, these mechanisms also affect plasticity, and thus learning within the cortex [Sjöström and Häusser, 2006, Ebner et al., 2019]. While the interplay between basal and apical stimulation and its effect on synaptic efficacies is subject to ongoing research, there is evidence that BAC-firing tends to shift plasticity towards long-term potentiation (LTP) [Letzkus et al., 2006]. Thus, coincidence between basal and apical input appears to also gate synaptic plasticity. Nonlinear Dendritic Coincidence Detection for Supervised Learning APREPRINT In a supervised learning scheme, where the top-down input arriving at the apical compartment acts as the teaching signal, the most straight-forward learning rule for the basal synaptic weights would be derived from an appropriate loss function, such as a mean square error, based on the difference between basal and apical input, i.e. Ip − Id, where indices p and d denote ‘proximal’ and ‘distal’, in equivalence to basal and apical. Theoretical studies have investigated possible learning mechanisms that could utilize an intracellular error signal [Urbanczik and Senn, 2014, Schiess et al., 2016, Guerguiev et al., 2017]. However, a clear experimental evidence for a physical quantity encoding such an error is—to our knowledge—yet to be found. On the other hand, Hebbian-type plasticity is extensively documented in experiments [Gustafsson et al., 1987, Debanne et al., 1994, Markram et al., 1997, Bi and Poo, 1998]. Therefore, our work is based on the question of whether the nonlinear interactions between basal and apical synaptic input could, when combined with a Hebbian plasticity rule, allow a neuron to learn to reproduce an apical teaching signal in its proximal input. We investigate coincidence learning by combining a phenomenological model that generates the output firing rate as a function of two streams of synaptic input (subsuming basal and apical inputs) with classical Hebbian, as well as BCM-like plasticity rules on basal synapses. In particular, we hypothesized that this combination of neural activation and plasticity rules would lead to an increased correlation between basal and apical inputs. Furthermore, the temporal alignment observed in our study could potentially facilitate apical inputs to act as top-down teaching signals, without the need for an explicit error-driven learning rule. Thus, we also test our model in a simple linear supervised classification task and compare it with the performance of a simple point neuron equipped with similar plasticity rules. 2 Model 2.1 Compartamental Neuron The neuron model used throughout this study is a discrete-time rate encoding model that contains two separate input variables, representing the total synaptic input current injected arriving at the basal (proximal) and apical (distal) dendritic structure of a pyramidal neuron, respectively. The model is a slightly simplified version of a phenomenological model proposed by Shai et al. [2015]. Denoting the input currents Ip (proximal) and Id (distal), the model is written as y (t) = ασ (Ip(t) − θp0) [1 − σ (Id(t) − θd)] (1) + σ (Id(t) − θd) σ (Ip(t) − θp1) 1 σ(x) ≡ : (2) 1 + exp(−4x) Here, θp0 > θp1 and θd are threshold variables with respect to proximal and distal inputs. Equation (1) defines the firing rate y as a function of Ip and Id. Note that the firing rate is normalized to take values within y 2 [0; 1]. In the publication by Shai et al. [2015], firing rates varied between 0 and 150 Hz. High firing rates typically appear in the form of bursts of action potentials, lasting on the order of 50–100 ms Larkum et al. [1999], Shai et al. [2015]. Therefore, since our model represents “instantaneous" firing rate responses to a discrete set of static input patterns, we conservatively estimate the time scale of our model to be on the order of tenths of seconds. In general, the input currents Ip and Id are meant to comprise both excitatory and potential inhibitory currents. Therefore, we did not restrict the sign of of Ip and Id to positive values. Moreover, since we chose the thresholds θp0 and θd to be zero, Ip and Id should be rather seen as a total external input relative to intrinsic firing thresholds. Note that the original form of this phenomenological model by Shai et al. [2015] is of the form y(Ip;Id) = σ (Ip − Aσ(Id)) [1 + Bσ(Id)] ; (3) where σ denotes the same sigmoidal activation function. This equation illustrates that Id has two effects: It shifts the basal activation threshold by a certain amount (here controlled by the parameter A) and also multiplicatively increases the maximal firing rate (to an extent controlled by B). Our equation mimics these effects by means of the two thresholds θp0 and θp1, as well as the value of α relative to the maximal value of y (which is 1 in our case). Overall, equation (1) describes two distinct regions of neural activation in the (Ip;Id)-space which differ in their maximal firing rates, which are set to 1 and α, where 0 < α < 1. A plot of (1) is shown in Fig. 1. When both input currents Id and Ip are large, that is, larger than the thresholds θd and θp1, the second term in (1) dominates, which leads to y ≈ 1. An intermediate activity plateau, of strength α emerges in addition when Ip > θp0 and Id < θd. As such, the compartment model (1) is able to distinguish neurons with a normal activity level, here encoded by α = 0:3, and strongly bursting neurons, where the maximal firing rate is unity. The intermediate plateau allows neurons to process the proximal inputs Ip even in the absence of distal stimulation. The distal current Id acts therefore as an additional modulator. 2 Nonlinear Dendritic Coincidence Detection for Supervised Learning APREPRINT Figure 1: Two-compartment rate model. The firing rate as a function of proximal and distal inputs Ip and Id, see (1). The thresholds θp0, θp1 and θd define two regions of neural activity, with a maximal firing rate of 1 and a plateau in the lower-left quadrant with a value of α = 0:3. That is, the latter region can achieve 30% of the maximal firing rate. In our numerical experiments, we compare the compartment model with a classical point neuron, as given by y(t) = σ (Ip(t) + Id(t) − θ) : (4) The apical input Id is generated ‘as is’, meaning it is not dynamically calculated as a superposition of multiple presynaptic inputs. For concreteness, we used Id(t) = nd(t)xd(t) − bd(t) ; (5) where nd(t) is a scaling factor, xd(t) a discrete time sequence, which represents the target signal to be predicted by the proximal input, and bd(t) a bias. In our experiments, we chose xd according to the prediction task at hand, see (17) and (19)–(20). Note that nd and bd are time dependent since they are subject to adaptation processes, which will be described in the next section.

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